Solution of Toeplitz normal equations by sine transform based preconditioning
type: Item
instance of: scholarly article
title: Solution of Toeplitz normal equations by sine transform based preconditioning
zbMATH Open document ID: 0935.65036
DOI: 10.1016/S0024-3795(98)10115-5
author: F. Di Benedetto
published in: Linear Algebra and its Applications
publication date: 1999-11-28T00:00:00Z
review text: The normal equations constructed by a Toeplitz matrix are studied and a suitable preconditioner related to the discrete sine transform is found. New results about the structure of the product of two Toeplitz matrices are given. This allows the conjugate gradient method applied to the normal equations to achieve a superlinear rate of convergence. The proposed preconditioner outperforms the circulant one for the iterative solution of Toeplitz least squares problems. Such a strategy can also be applied to linear systems with nonsymmetric coefficient matrices. The block generalization of the algorithm is discussed which make the new technique applicable to some problems in multichannel signal processing and two-dimensional image restoration. Results and discussion from numerical experiments using MATLAB are given.
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 68U10
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 1371024
zbMATH Keywords: Toeplitz matrix
zbMATH Keywords: discrete sine transform
zbMATH Keywords: normal equations
zbMATH Keywords: preconditioning
zbMATH Keywords: conjugate gradient method
zbMATH Keywords: convergence
zbMATH Keywords: least squares problems
zbMATH Keywords: algorithm
zbMATH Keywords: signal processing
zbMATH Keywords: image restoration
zbMATH Keywords: numerical experiments
reviewed by: Krassimir Georgiev
describes a project that uses: Matlab
describes a project that uses: CGS
MaRDI profile type: Publication
cites work: Superfast Solution of Real Positive Definite Toeplitz Systems
cites work: Spectral and computational properties of band symmetric Toeplitz matrices
cites work: Fast Transform Based Preconditioners for Toeplitz Equations
cites work: Stability of Methods for Solving Toeplitz Systems of Equations
cites work: A Family of Block Preconditioners for Block Systems
cites work: FFT-Based Preconditioners for Toeplitz-Block Least Squares Problems
cites work: Circulant Preconditioned Toeplitz Least Squares Iterations
cites work: Displacement preconditioner for Toeplitz least squares iterations
cites work: Toeplitz preconditioners for Hermitian Toeplitz systems
cites work: Conjugate Gradient Methods for Toeplitz Systems
cites work: Sine transform based preconditioners for symmetric Toeplitz systems
cites work: Toeplitz Equations by Conjugate Gradients with Circulant Preconditioner
cites work: An Optimal Circulant Preconditioner for Toeplitz Systems
cites work: Analysis of Preconditioning Techniques for Ill-Conditioned Toeplitz Matrices
cites work: Preconditioning of Block Toeplitz Matrices by Sine Transforms
cites work: C. G. preconditioning for Toeplitz matrices
cites work: A unifying approach to abstract matrix algebra preconditioning
cites work: Q5185900
cites work: Methods of conjugate gradients for solving linear systems
cites work: On the Spectrum of a Family of Preconditioned Block Toeplitz Matrices
cites work: Spectral Properties of Preconditioned Rational Toeplitz Matrices: The Nonsymmetric Case
cites work: A special class of structured matrices constructed with the Kronecker product and its use for difference equations
cites work: On the distribution of the singular values of Toeplitz matrices
cites work: Superlinear PCG methods for symmetric Toeplitz systems
cites work: CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems
cites work: Influence of matrix operations on the distribution of eigenvalues and singular values of Toeplitz matrices
cites work: A unifying approach to some old and new theorems on distribution and clustering
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description: scientific article; zbMATH DE number 1371024